The two sides had been in talks over a deal for the playmaker worth £52.75m plus a maximum of £4.4m in add-ons but he will be staying with the Ligue 1 club.
However, it would appear Lyon president Jean-Michel Aulas is likely to sell Fekir after the World Cup.
Liverpool had previously been expected to announce the signing of Fekir on Friday night.
"Although, Liverpool had been the priority of a possible transfer for Nabil, and subject to other proposals consistent with the interest of Nabil and Olympique Lyonnais and from ambitious clubs, Olympique Lyonnaise is delighted to be able to count on the presence of our captain, who is a leading player for the 2018/19 season when the club will play in the Champions League".
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He came through Lyon's youth system and has made 153 appearances for the club, scoring 57 goals, having made his debut in 2013.
Le Graet, speaking in Lyon a few hours before France played the United States in a World Cup warm up game in the city, said: "These are negotiations".
The collapse - for now at least - of Liverpool's move for Fekir will be a setback for Jurgen Klopp.
France forward Fekir has always been linked with a big-money move to Anfield and enjoyed a stellar campaign in France's top flight this season - scoring 18 goals and supplying a further eight assists.
However, reports in France suggested the delay may be down to Lyon's accountancy practices wanting the deal to be completed in the new financial year.